Chasing 121 for victory, the home side achieved the target in the final day's morning session with KL Rahul making 58 not out and B Sai Sudharsan contributing 39.
"It's a really big honour and I would say I'm getting used to it," said home captain Shubman Gill, who registered his first series victory since taking over as Test skipper earlier this year.
"Managing all the players and leading this team is a great honour."
India won the series opener in Ahmedabad inside three days and put themselves in the box seat in the second match when they amassed 518-5 before declaring their first innings.
West Indies folded for 248 in the first innings and were made to follow on.
The visitors delivered a much-improved batting display in their second innings and rode hundreds by John Campbell and Shai Hope to post 390 all out to stretch the contest to its final day.
Resuming on 63-1, India needed just an hour to complete the chase, but West Indies managed to grab a couple of wickets courtesy of two superb catches.
Touring captain Roston Chase dismissed Sudharsan with Hope diving to take a low catch in the slip.
Gill (13) hit a six and a four before miscuing a ball from Rose. Justin Greaves locked onto the swirling ball and ran from midwicket to take a brilliant catch.
That was the last of the drama, and Rahul hit the winning boundary to seal India's win.
India spinner Kuldeep Yadav, who claimed a match haul of eight wickets, was named player of the match and teammate Ravindra Jadeja player of the tournament.
West Indies captain Chase said his side would take some positives from the defeat.
"The positives for us in this match were that Campbell and Hope played well and scored hundreds," he added.
"We batted 100 overs after a long time; that was another positive. Taking the game to the fifth day, that was great for us."
